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Lemma 4.1.2. Let P₁, P2, P3 be three arcs, between the same two end- points but otherwise disjoint. (i) R² (P₁ UP₂ U P3) has exactly three regions, with frontiers P₁UP2, P2UP3 and P₁ UP3. (ii) If P is an arc between a point in P₁ and a point in P3 whose interior lies in the region of R2 (P₁ UP3) that contains P2, then PnP₂ # 0.

Lemma 4.1.2. Let P₁, P2, P3 be three arcs, between the same two end- points but otherwise disjoint. (i) R² (P₁ UP₂ U P3) has exactly three regions, with frontiers P₁UP2, P2UP3 and P₁ UP3. (ii) If P is an arc between a point in P₁ and a point in P3 whose interior lies in the region of R2 (P₁ UP3) that contains P2, then PnP₂ # 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Lemma 4.1.2. Let P₁, P2, P3 be three arcs, between the same two end- points but otherwise disjoint. (i) R² (P₁ UP₂ U P3) has exactly three regions, with frontiers P₁UP2, P2UP3 and P₁ UP3. (ii) If P is an arc between a point in P₁ and a point in P3 whose interior lies in the region of R2 (P₁UP3) that contains P2, then POP₂ # 0. P1 P2 P3 Fig. 4.1.1. The arcs in Lemma 4.1.2 (ii)
Transcribed Image Text:Lemma 4.1.2. Let P₁, P2, P3 be three arcs, between the same two end- points but otherwise disjoint. (i) R² (P₁ UP₂ U P3) has exactly three regions, with frontiers P₁UP2, P2UP3 and P₁ UP3. (ii) If P is an arc between a point in P₁ and a point in P3 whose interior lies in the region of R2 (P₁UP3) that contains P2, then POP₂ # 0. P1 P2 P3 Fig. 4.1.1. The arcs in Lemma 4.1.2 (ii)
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